For a \(K\) -linear map \(f : V \rightarrow W\) , the kernel of \(f\)  is the equivalence
relation \(\mathrm {ker}(f)= \lbrace (v_1,v_2) \mid f(v_1) = f(v_2)\rbrace \) . It can be shown
that \(\mathrm {ker}(f)\)  is a congruence. It follows that \(V/\mathrm {ker}(f)\)  is a
semimodule. Further, every \(K\) -linear map \(f\)  has a unique (up to
isomorphism) factorization into two \(K\) -linear maps, a surjection \(\varepsilon _f: V \rightarrow V/\mathrm {ker}(f)\)  followed by an injection \(\mu _f: V/\mathrm {ker}(f) \rightarrow W\)  defined
by \(\varepsilon _f(u) = [u]\)  and \(\mu _f([u])=f(u)\) . In the following, when we refer to the kernel
of a \(K\) -linear map \(f\) , we mean the kernel of \(f\)  as defined above and and
not the set \(\lbrace u \mid f(u)=0\rbrace \) . This usage is common in universal
algebra [1]}.
